28.1.90 problem 111

Internal problem ID [4396]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 2. First-Order and Simple Higher-Order Differential Equations. Page 78
Problem number : 111
Date solved : Sunday, March 30, 2025 at 03:16:41 AM
CAS classification : [[_homogeneous, `class G`], _rational, _Bernoulli]

\begin{align*} x y^{2} \left (x y^{\prime }+y\right )&=1 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 72

ode:=x*y(x)^2*(y(x)+x*diff(y(x),x)) = 1; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {\left (12 x^{2}+8 c_1 \right )^{{1}/{3}}}{2 x} \\ y &= -\frac {\left (12 x^{2}+8 c_1 \right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{4 x} \\ y &= \frac {\left (12 x^{2}+8 c_1 \right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{4 x} \\ \end{align*}

✓ Mathematica. Time used: 0.244 (sec). Leaf size: 80

ode=x*y[x]^2*(x*D[y[x],x]+y[x])==1; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {\sqrt [3]{-\frac {1}{2}} \sqrt [3]{3 x^2+2 c_1}}{x} \\ y(x)\to \frac {\sqrt [3]{\frac {3 x^2}{2}+c_1}}{x} \\ y(x)\to \frac {(-1)^{2/3} \sqrt [3]{\frac {3 x^2}{2}+c_1}}{x} \\ \end{align*}

✓ Sympy. Time used: 1.630 (sec). Leaf size: 80

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(x*Derivative(y(x), x) + y(x))*y(x)**2 - 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \frac {2^{\frac {2}{3}} \sqrt [3]{\frac {\frac {C_{1}}{x^{2}} + 3}{x}}}{2}, \ y{\left (x \right )} = \frac {2^{\frac {2}{3}} \sqrt [3]{\frac {\frac {C_{1}}{x^{2}} + 3}{x}} \left (-1 - \sqrt {3} i\right )}{4}, \ y{\left (x \right )} = \frac {2^{\frac {2}{3}} \sqrt [3]{\frac {\frac {C_{1}}{x^{2}} + 3}{x}} \left (-1 + \sqrt {3} i\right )}{4}\right ] \]